You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
16 lines
1.1 KiB
16 lines
1.1 KiB
\emph{A formula $\varphi$ is valid, if and only if, $\neg \varphi$ is not satisfiable}.
|
|
|
|
Consider the formula $\varphi=(x \lor \neg x)$. This formula is valid, i.e., all rows in the truth table
|
|
would evaluate to \emph{true}. The negation of $\varphi$ is the following:
|
|
$\neg \varphi = \neg (x \lor \neg x) = \neg x \land x$, which is not satisfiable, i.e., all rows the truth table would evaluate to \emph{false}.
|
|
|
|
\noindent\emph{A formula $\varphi$ is satisfiable, if and only if, $\neg \varphi$ is not valid}.
|
|
|
|
If $\varphi$ is satisfiable, there is at least one model that makes the formula true. If we negate the formula,
|
|
these models make the negated formula false, and therefore, the negated formula cannot be valid.
|
|
Consider the formula $\varphi=(x \lor y)$.
|
|
There is at least one model that makes the formula true, e.g. $\mathcal{M} \coloneqq x = T, \ y = T$.
|
|
The negation of $\varphi$ is the following:
|
|
$\neg \varphi = \neg (x \lor y) = \neg x \land \neg y$.
|
|
Under the same model $\mathcal{M}$ as before, $\neg \varphi$ evaluates to false.
|
|
So the negated formula is not valid.
|